When the pair temperature is even higher, i.e., \(T_p\sim \textrm {MeV}\) , the spectrum shows new features, as shown in Fig. REF  for one example with \(T_p=1 \ \mathrm {MeV}\) . Broadband plasma waves are excited due to the relativistic thermal spread of the particle beams. The \(n\omega _{pe}\) -radiation peaks are completely smeared out. In addition, the simulation shows the thermal expansion dominates the interaction, leading to a hot pair-dominated plasma in front of the target, consistent with previous studies [1]}, [2]}. The laser coupling with the pair plasmas induces high-frequency modulations on the reflected laser field. These modulations enhance the power of high frequency harmonics (\(\gtrsim 3\omega _{pe}\) ) in the spectrum. This results in a broadband plateau in the spectrum. The height of the plateau increases with \(T_p\) , explaining the rise of radiation power at \(2\omega _{pe}\)  in the \(\sim \textrm {MeV}\)  regime in Fig. REF (a). We note that the spectral features here with \(\sim \ \mathrm {MeV}\)  pair plasma are also consistent with the self-consistent scenario studied in Sec. , where denser and hotter pair plasmas are created in the interaction region.
